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We consider the energy-critical defocusing nonlinear wave equation on$\mathbb{R}^4$ and establish almost sure global existence and scattering forrandomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \timesH^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almostsure scattering result for an energy-critical dispersive or hyperbolic equationwith scaling super-critical initial data. The proof is based on theintroduction of an approximate Morawetz estimate to the random data setting andnew large deviation estimates for the free wave evolution of randomizedradially symmetric data.

Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data